CAREER: Phase Transitions in Some Discrete Random Models and Mixing of Markov Chains
职业:一些离散随机模型中的相变和马尔可夫链的混合
基本信息
- 批准号:1554783
- 负责人:
- 金额:$ 40万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2016
- 资助国家:美国
- 起止时间:2016-07-01 至 2023-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Water turning into ice at its freezing point or the magnetization of iron are examples of phase transitions in physical systems. At the transition point, the properties of the system such as the volume or heat capacity may change discontinuously. The aim of our research is to study phase transitions in mathematical models using probabilistic tools in the following three directions. (1) The longest increasing subsequence (LIS) of a permutation is the length of a maximal subsequence of the permutation in which the elements increase. How long is the LIS for a uniformly random permutation? This question has been studied in connection with practical applications such as sorting sequences, disk drive scheduling and airplane boarding times. The mathematical study of the LIS has revealed deep and unexpected connections of the problem with areas such as the theory of random matrices, analytic combinatorics and random polymer models. The proposed research aims to study the LIS when the permutation is drawn from certain non-uniform distributions and associated phase transitions. (2) Many computational problems can be phrased as constraint satisfaction problems (CSPs) where one wants to find a solution to a number of variables with a set of constraints imposed on them. CSPs were first studied in computer science motivated by applications to artificial intelligence. To study the difficulty of finding solutions in typical rather than worst case scenarios, researchers study random CSPs. Using sophisticated heuristics, physicists have made detailed predictions about the location and nature of phase transitions in random CSPs. The accuracy of these heuristic predictions motivates the importance of discovering the rigorous mathematical foundations of these techniques. (3) Interacting particle processes are used to model large, randomly evolving interacting systems of agents that arise in the natural sciences including in physics and in biology. The exclusion and interchange random walks are examples of such interacting particle processes. In the symmetric case the long term mixing behavior of the random walk and the nature of phase transitions is well studied. The goal of this research is to understand the mixing properties of natural asymmetric and weighted versions of these processes. While achieving these three goals, the principal investigator will create exciting research opportunities for graduate and undergraduate students in probability, mentoring programs with the goal of retention of women in mathematics, and the development of online curricular material.The main aim of this project is to develop new theory and analysis for phase transitions in certain discrete probabilistic models. The first problem is to study the limiting distribution of the LIS in non-uniformly random permutations by way of analyzing the fluctuations of the LIS as the parameter of the distribution is varied. The distribution is known to be Gaussian in one regime of the parameter and Tracy-Widom in another and we aim to study this transition. The second problem is to study the condensation and clustering transitions in random CSPs such as the hardcore model on random graphs. The research aims to identify the location of the reconstruction threshold more precisely in these models and to explore the connection to the clustering transition. Finally, the proposal will consider Markov processes such as asymmetric exclusion and interchange and attempt to relate the mixing times and spectral gaps of these processes to the corresponding quantities for a single particle and to understand the cutoff phenomenon for these processes.
水在冰点变成冰或铁的磁化是物理系统中相变的例子。在转变点,系统的性质,如体积或热容,可能会不连续地变化。我们研究的目的是在以下三个方向使用概率工具研究数学模型中的相变。(1)一个置换的最长递增子序列(LIS)是其中元素增加的置换的最大子序列的长度。均匀随机排列的LIS有多长?这个问题已经研究了实际应用,如排序序列,磁盘驱动器调度和飞机登机时间。LIS的数学研究揭示了这个问题与随机矩阵理论、分析组合学和随机聚合物模型等领域的深刻而意想不到的联系。 拟议的研究旨在研究LIS时的排列是从某些非均匀分布和相关的相变。(2)许多计算问题可以被称为约束满足问题(CSP),其中人们希望找到一个解决方案,以解决一些变量与一组约束强加给他们。CSP最初是在计算机科学中被人工智能应用所激发的。为了研究在典型而非最坏情况下找到解决方案的难度,研究人员研究了随机CSP。利用复杂的物理学,物理学家对随机CSP中相变的位置和性质做出了详细的预测。这些启发式预测的准确性激发了发现这些技术的严格数学基础的重要性。(3)相互作用粒子过程被用来模拟大的,随机演变的相互作用系统的代理人出现在自然科学,包括在物理学和生物学。排斥随机游动和交换随机游动就是这种相互作用粒子过程的例子。在对称情况下,随机游动的长期混合行为和相变的性质得到了很好的研究。本研究的目的是了解这些过程的自然不对称和加权版本的混合特性。在实现这三个目标的同时,首席研究员将为概率学领域的研究生和本科生创造令人兴奋的研究机会,开展旨在留住女性数学人才的指导计划,以及开发在线课程材料。该项目的主要目的是开发新的理论和分析某些离散概率模型中的相变。第一个问题是通过分析LIS随分布参数变化的波动来研究LIS在非均匀随机排列中的极限分布。已知该分布在参数的一个区域中是高斯分布,而在另一个区域中是Tracy-Widom分布,我们的目标是研究这种转变。第二个问题是研究随机CSP(如随机图上的硬核模型)中的凝聚和聚类转移。该研究旨在更精确地识别这些模型中重建阈值的位置,并探索与聚类过渡的联系。最后,该提案将考虑马尔可夫过程,如不对称排斥和交换,并试图将这些过程的混合时间和光谱间隙与单个粒子的相应量联系起来,并理解这些过程的截止现象。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Naya Banerjee其他文献
Naya Banerjee的其他文献
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{{ truncateString('Naya Banerjee', 18)}}的其他基金
Probability Applied to Problems in Algorithmic Statistics, Statistical Physics and the Combinatorics of Permutations
概率应用于算法统计、统计物理和排列组合问题
- 批准号:
1261010 - 财政年份:2012
- 资助金额:
$ 40万 - 项目类别:
Standard Grant
Probability Applied to Problems in Algorithmic Statistics, Statistical Physics and the Combinatorics of Permutations
概率应用于算法统计、统计物理和排列组合问题
- 批准号:
1208348 - 财政年份:2012
- 资助金额:
$ 40万 - 项目类别:
Standard Grant
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